Julia's Efficient Algorithm for Subtyping Unions and Covariant Tuples
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Julia’s efficient algorithm for subtyping unions and covariant tuples (Pearl) Benjamin Chung, Francesco Zappa Nardelli, Jan Vitek To cite this version: Benjamin Chung, Francesco Zappa Nardelli, Jan Vitek. Julia’s efficient algorithm for subtyping unions and covariant tuples (Pearl). ECOOP 2019 - 33rd European Conference of Object-Oriented Programming, Jul 2019, London, United Kingdom. 10.4230/LIPIcs.ECOOP.2019.6. hal-02297696 HAL Id: hal-02297696 https://hal.inria.fr/hal-02297696 Submitted on 26 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 Julia’s efficient algorithm for subtyping unions and 2 covariant tuples (Pearl) 3 Benjamin Chung 4 Northeastern University 5 Francesco Zappa Nardelli 6 Inria 7 Jan Vitek 8 Northeastern University & Czech Technical University in Prague 9 Abstract 10 The Julia programming language supports multiple dispatch and provides a rich type annotation 11 language to specify method applicability. When multiple methods are applicable for a given call, 12 Julia relies on subtyping between method signatures to pick the correct method to invoke. Julia’s 13 subtyping algorithm is surprisingly complex, and determining whether it is correct remains an open 14 question. In this paper, we focus on one piece of this problem: the interaction between union 15 types and covariant tuples. Previous work normalized unions inside tuples to disjunctive normal 16 form. However, this strategy has two drawbacks: complex type signatures induce space explosion, 17 and interference between normalization and other features of Julia’s type system. In this paper, 18 we describe the algorithm that Julia uses to compute subtyping between tuples and unions—an 19 algorithm that is immune to space explosion and plays well with other features of the language. We 20 prove this algorithm correct and complete against a semantic-subtyping denotational model in Coq. 21 2012 ACM Subject Classification Theory of computation → Type theory 22 Keywords and phrases Type systems, Subtyping, Union types 23 Digital Object Identifier 10.4230/LIPIcs.ECOOP.2019.6 24 Category Pearl 25 1 Introduction 26 Union types, originally introduced by Barbanera and Dezani-Ciancaglini [2], are being 27 adopted in mainstream languages. In some cases, such as Julia [5] or TypeScript [11], they 28 are exposed at the source level. In others, such as Hack [8], they are only used internally as 29 part of type inference. As a result, subtyping algorithms between union types are of increasing 30 practical import. The standard subtyping algorithm for this combination of features has, for 31 some time, been exponential in both time and space. An alternative algorithm, linear in space 32 but still exponential in time, has been tribal knowledge in the subtyping community [15]. In 33 this paper, we describe and prove correct an implementation of that algorithm. 34 We observed the algorithm in our prior work formalizing the Julia subtyping relation [17]. 35 There, we described Julia’s subtyping relation as it arose from its decision procedure but were 36 unable to prove it correct. Indeed, we found bugs in the Julia implementation and identified 37 unresolved correctness issues. Contemporary work addresses some correctness concerns [3] 38 but leaves algorithmic correctness open. 39 Julia’s subtyping algorithm [4] is used for method dispatch. While Julia is dynamically 40 typed, method arguments can have type annotations. These annotations allow one method 41 to be implemented by multiple functions. At run time, Julia searches for the most specific 42 applicable function for a given invocation. Consider these declarations of multiplication: 43 © Benjamin Chung, Francesco Zappa Nardelli, Jan Vitek; licensed under Creative Commons License CC-BY 33rd European Conference on Object-Oriented Programming (ECOOP 2019). Editor: Alastair F. Donaldson; Article No. 6; pp. 6:1–6:15 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 6:2 Subtyping union types and covariant tuples 44 45 *(x::Number, r::Range) = range(x*first(r),...) 46 *(x::Number, y::Number) = *(promote(x,y)...) 47 48 *(x::T, y::T) where T <: Union{Signed,Unsigned} = mul_int(x,y) 49 The first two methods implement, respectively, multiplicaton of a range by a number and 50 generic numeric multiplication. The third method invokes native multiplication when both 51 arguments are either signed or unsigned integers (but not a mix of the two). Julia uses 52 subtyping to decide which of the methods to call at any specific site. The call 1*(1:4) 53 dispatches to the first, 1*1.1 the second, and 1*1 the third. 54 Julia offers programmers a rich type language to express complex relationships in type 55 signatures. The type language includes nominal primitive types, union types, existential 56 types, covariant tuples, invariant parametric datatypes, and singletons. Intuitively, subtyping 57 between types is based on semantic subtyping: the subtyping relation between types holds 58 when the sets of values they denote are subsets of one another [5]. We write the set of values 59 represented by a type t as t . Under semantic subtyping, the types t1 and t2 are subtypes J K 60 iff t1 ⊆ t2 . From this, we derive a forall-exists intuition for subtyping: for every value J K J K 61 denoted on the left-hand side, there must exist some value on the right-hand side to match 62 it, thereby establishing the subset relation. This simple intuition is, however, complicated to 63 check algorithmically. 64 In this paper, we focus on the interaction of two features: covariant tuples and union 65 types. These two kinds of type are important to Julia’s semantics. Julia does not record 66 return types, so a function’s signature consists solely of the tuple of its argument types. 67 These tuples are covariant, as a function with more specific arguments is preferred to a more 68 generic one. Union types are widely used as shorthand to avoid writing multiple functions 69 with the same body. As a consequence, Julia library developers write many functions with 70 union typed arguments, functions whose relative specificity must be decided using subtyping. 71 To prove the correctness of the subtyping algorithm, we first examine typical approaches 72 in the presence of union types. Based on Vouillon [16], the following is a typical deductive 73 system for subtyping union types: allexist existL existR tuple 0 00 0 00 0 0 ft <: t t <: t t <: t t <: t t1 <: t1 t2 <: t2 0 00 0 00 0 00 0 0 Union{t , t } <: t t <: Union{t , t } t <: Union{t , t } Tuple{t1, t2} <: Tuple{t1, t2} 7574 While this rule system might seem to make intuitive sense, it does not match the semantic 76 intuition for subtyping. For instance, consider the following judgment: 0 00 0 00 77 Tuple{Union{t , t }, t} <: Union{Tuple{t , t}, Tuple{t , t}} 79 Using semantic subtyping, the judgment should hold. The set of values denoted by a 78 80 union Union{t1, t2} is just the union of the set of values denoted by each of its members J K 81 t1 ∪ t2 . A tuple Tuple{t1, t2}’s denotation is the set of tuples of the respective values J K J K 82 {Tuple{v1, v2} | v1 ∈ t1 ∧ v2 ∈ t2 }. Therefore, the left-hand side denotes the values 0 00 0 J0 K 00 00J K 0 83 {Tuple{v , v } | v ∈ t ∪ t ∧ v ∈ t }, while the right-hand side denotes Tuple{t , t} ∪ 00 J K J K J 0K 00 0 0 00 00 J K 84 Tuple{t , t} or equivalently {Tuple{v , v } | v ∈ t ∪ t ∧ v ∈ t }. These sets are the J K J K J K J K 85 same, and therefore subtyping should hold in either direction between the left- and right-hand 86 types. However, we cannot derive this relation from the above rules. According to them, we 0 00 87 must pick either t or t on the right-hand side using existL or existR, respectively, ending 0 00 0 0 00 00 88 up with either Tuple{Union{t , t }, t} <: Tuple{t , t} or Tuple{Union{t , t }, t} <: Tuple{t , t}. 89 In either case, the judgment does not hold. How can this problem be solved? 90 Most prior work addresses this problem by normalization [2, 14, 1], rewriting all types into 91 their disjunctive normal form, as unions of union-free types, before building the derivation. B. Chung, F. Zappa Nardelli, J. Vitek 6:3 92 Now all choices are made at the top level, avoiding the structural entanglements that cause 93 difficulties. The correctness of this rewriting step comes from the semantic denotational 94 model, and the resulting subtyping algorithm can be proved both correct and complete. Other 95 proposals, such as Vouillon [16] and Dunfield [7], do not handle distributivity. Normalization 96 is used by Frisch et al.’s [9], by Pearce’s flow-typing algorithm [13], and by Muehlboeck 97 and Tate in their general framework for union and intersection types [12]. Few alternatives 98 have been proposed, with one example being Damm’s reduction of subtyping to regular tree 99 expression inclusion [6]. 100 However, a normalization-based algorithm has two major drawbacks: it is not space 101 efficient, and other features of Julia render it incorrect.